(i) Skill Extraction from Offline Data:
We define a continuous skill \(z_i \in \mathcal {Z}\)  as an embedding for a sequence of state-action pairs \(\lbrace s_t, a_t, \dots , s_{t+H-1}, a_{t+H-1}\rbrace \)  with a fixed length H.
This temporal abstraction of skills has proven to be useful in prior work [1]}, [2]}, by allowing a hierarchical decomposition of skills to achieve long horizon downstream tasks.
To learn the latent space \(\mathcal {Z}\)  we propose training a continuous latent variable model with the encoder as \(q_{\phi }(z \vert s_t, a_t, \dots , s_{t+H-1}, a_{t+H-1})\)  and the decoder as \(\pi _{\theta }(a \vert s, z)\) .
The encoder outputs a distribution over the latent variable \(z\)  that best explains the variation in the state-action pairs in the sub-trajectory.
